"The back end of the bullpen is an issue that has to be dealt with," Astros general manager Ed Wade said.
Hawkins and Valverde are both Type A free agents, meaning the Astros would get compensatory Draft picks in return if Houston offers them arbitration and they sign with other teams. Valverde made $8 million last season and has said he wants to test the free-agent waters, and Hawkins made $3.5 million.
The Astros have been in contact with the agents for both pitchers and would like to re-sign both. Considering Houston's payroll is going to shrink next year, retaining both could be a challenge.
"Whether being able to retain Valverde or do something in free agency or make a trade or take a younger guy and try to push him to the back end of the bullpen, we have to be open-minded," Wade said.
Valverde, 31, posted the second-lowest ERA of his career at 2.33 while appearing in 52 games. He was 25-for-29 in save chances and went 4-1 with 23 saves and a 1.76 ERA in 44 games after returning from a lengthy stay on the disabled list with a strained calf.
Valverde is one of several big-name closers on the market, a list that includes Fernando Rodney, Kevin Gregg and Billy Wagner.
"There's a lot of fluidity on the market," Wade said. "There could be some other guys become available depending on their arbitration status."
Hawkins, 36, posted a club-best 2.13 ERA and had 11 saves, converting seven saves while Valverde was on the disabled list. He blew four saves, but the Astros won all four of those games. His 11 saves were the most since he had 25 with the Cubs in 2004.
"I think Hawk has really established himself since arriving on the scene here as a guy we can count on late in games, whether it's the eighth or the ninth inning," Wade said. "I know there's a magic to the ninth inning and there's no walking past that. He does have ninth-inning experience.
"He's done it before, and I think the way Hawk is throwing since we made the trade [with the Yankees] in July of '08 indicates to me this is a guy that you can count on in the eighth and ninth inning and has the stuff to do it. That said, first and foremost we've got to get him back here and then address things as we go forward."
Astros rookie Jeff Fulchino proved last year he was capable of pitching late in games. He went 6-4 with a 3.40 ERA in 61 relief appearances en route to being named as the Astros' Rookie of the Year by the Houston chapter of the Baseball Writers' Association of America.
Brian McTaggart is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.